Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 25, 1-19; http://www.math.u-szeged.hu/ejqtde/

Oscillation of Second-Order Nonlinear Differential Equations with Damping Term

E.M.Elabbasy^{1} and W.W.Elhaddad Department of Mathematics Faculty of Science Mansoura University Mansoura, 35516, Egypt 1E-mail: emelabbasy@mans.edu.eg

Abstract

We present new oscillation criteria for the second order nonlinear differential equation with damping term of the form

(r(t)ψ(x)f(ẋ))^{·} + p(t)φ (g(x), r(t)ψ(x)f(ẋ)) + q(t)g(x) = 0,

where p, q, r : [t_{o}, ∞) → R and ψ, g, f : R → R are continuous, r(t) > 0, p(t) ≥ 0 and ψ(x) > 0, xg(x) > 0 for x ≠ 0, uf(u) > 0 for u ≠ 0. Our results generalize and extend some known oscillation criteria in the literature. The relevance of our results is illustrated with a number of examples.

1 Introduction

We are concerned with the oscillation of solutions of second order differential equations with damping terms of the following form

(r(t)ψ(x)f(ẋ))^{·} + p(t)φ (g(x), r(t)ψ(x)f(ẋ)) + q(t)g(x) = 0 (· = \frac{d}{dt}) (E)

where r ∈ C[[t_{o}, ∞), R^{+}], p ∈ C[[t_{o}, ∞), [0, ∞)], q ∈ C[[t_{o}, ∞), R], ψ ∈ C[R, R^{+}] and g ∈ C^{1}[R, R] such that xg(x) > 0 for x ≠ 0 and g′(x) > 0 for x ≠ 0. φ is defined and continuous on R × R − {0} with uφ(u, v) > 0 for uv ≠ 0 and φ(λu, λv) = λφ(u, v) for 0 < λ < ∞ and (u, v) ∈ R × R − {0}.

We recall that a function x : [t_{o}, t_{1}) → R is called a solution of equation (E) if x(t) satisfies equation (E) for all t ∈ [t_{o}, t_{1}). In the sequel it will be always assumed that solutions of equation (E) exist for any t_{o} ≥ 0. Such a solution x(t) is said to be oscillatory if it has arbitrarily large zeros, and otherwise it said to be nonoscillatory. Equation (E) is called oscillatory if all its solutions are oscillatory.

Oscillatory and nonoscillatory behavior of solutions for various classes of second-order differential equations have been widely discussed in the literature (see, for example, [1 − 37] and the references quoted therein). There is a great number of papers dealing with particular cases of equation (E) such as the linear equations

ẍ(t) + q(t)x(t) = 0,  (E_{1})
(r(t)ẋ(t))^{·} + q(t)x(t) = 0,  (E_{2})

the nonlinear equations

ẍ(t) + q(t)g (x) = 0,  (E_{3})
(r(t)ψ(x)ẋ(t))^{·} + q(t)g (x) = 0,  (E_{4})

and the nonlinear equations with damping term

(r(t)ẋ(t))^{·} + p(t)ẋ(t) + q(t)g (x) = 0,  (E_{5})
(r(t)ψ(x)ẋ(t))^{·} + p(t)ẋ(t) + q(t)g (x) = 0,  (E_{6})
(r(t)f(ẋ))^{·} + p(t)φ (g(x), r(t)f(ẋ)) + q(t)g(x) = 0.  (E_{7})

An important tool in the study of oscillatory behaviour of solutions for equations (E_{1})−(E_{7}) is the averaging technique. This goes back as far as the classical results of Wintner [32] which proved that (E_{1}) is oscillatory if

lim_{t→∞} \frac{1}{t} ∫_{t_{o}}^{t} ∫_{t_{o}}^{s} q(u)du ds = ∞,

and Hartman [10] who showed the above limit cannot be replaced by the limit superior and proved the condition

lim inf_{t→∞} \frac{1}{t} ∫_{t_{o}}^{t} ∫_{t_{o}}^{s} q(u)du ds < lim sup_{t→∞} \frac{1}{t} ∫_{t_{o}}^{t} ∫_{t_{o}}^{s} q(u)du ds ≤ ∞,

implies that equation (E_{1}) is oscillatory.

The result of Wintner was improved by Kamenev [11] who proved that the condition

lim sup_{t→∞} \frac{1}{t^{n−1}} ∫_{t_{o}}^{t}(t − s)^{n−1}q(s)ds = ∞ for some n > 2,

is sufficient for the oscillation of equation (E_{1}).

Some other results can be found in [21], [27], [34] and the references therein. Kong [12] and Li [15] employed the technique of Philos [27] and obtained several oscillation results for (E_{2}).

Butler [2], Philos [22], [23], [24], [25], [26], Philos and Purnaras [28], Wong and Yeh [33] and Yeh [37] obtained some sufficient conditions for the oscillation of equation (E_{3}) and Grace [7], Elabbasy [3], Manojlovic [18] and Tiryaki and Cakmak [31] for the oscillation of equation (E_{4}).

In the presence of damping, a number of oscillation criteria have been obtained by Li and Agarwal [16], Grace and Lalli [9], Rogovchenko [29], Kirane and Rogovchenko [14], Li et al. [17], Yang [36] Nagabuchi and Minora Yamamoto [20], Yan [35], Elabbasy, Hassan and Saker [5] for equation (E_{5}), Ayanlar and Tiryaki [1], Tiryaki and Zafer [30], Kirane and Rogovchenko [13], Grace [6], [7],

[8] and Manojlovic [19] for equation (E_{6}) and Elabbasy and Elsharabasy [4] for equation (E_{7}).

In this paper we extend the results of Wintner [32], Yan [35], Elabbasy [3], Elabbasy and Elsharabasy [4], and Nagabuchi and Yamamoto [20] for a broad class of second order nonlinear equation of the type (E).

2 Main Results

Theorem 1. Suppose, in addition to condition

φ(1, z) ≥ z for all z ≠ 0,  (1)
\frac{g′(x)}{ψ(x)} ≥ K > 0 for all x ≠ 0,  (2)

∫_{±ε}^{±∞} \frac{ψ(y)}{g(y)}dy < ∞ for all ε > 0,  (3)

and

0 < L_{1} ≤ \frac{f(y)}{y} ≤ L_{2} for all y ≠ 0 and lim_{y→∞} \frac{f(y)}{y} exist,  (4)

that there exist a positive function ρ ∈ C^{1}[t_{o}, ∞) such that (ρ(t)r(t))^{·} ≤ 0 for all t ≥ t_{o}. Then equation (E) is oscillatory if

lim_{t→∞} sup \frac{1}{t} ∫_{t_{o}}^{t} ∫_{t_{o}}^{s} [ρ(u)q(u) − \frac{r(u)ρ(u)}{4M} (\frac{ρ̇(u)}{ρ(u)} − p(u))^{2}] duds = ∞, (5)

where M = \frac{K}{L_{2}} . Proof. On the contrary we assume that (E) has a nonoscillatory solution x(t). We suppose without loss of generality that x(t) > 0 for all t ∈ [t_{o}, ∞). We define the function ω(t) as

ω(t) = ρ(t)\frac{r(t)ψ(x)f(ẋ)}{g(x(t))} for all t ≥ t_{o}.

This and equation (E) imply

ω̇(t) = \frac{ρ̇(t)}{ρ(t)}ω(t) − ρ(t)[q(t) + p(t)φ(1, \frac{ω(t)}{ρ(t)})] − ρ(t)\frac{r(t)ψ(x)f(ẋ)g′(x)ẋ}{g^{2}(x(t))}.

From (1), (2) and (4) we obtain

ω̇(t) ≤ −ρ(t)q(t) − (\frac{ρ̇(t)}{ρ(t)} − p(t)) ω(t) − \frac{M}{ρ(t)r(t)}ω^{2}(t).

Integrating from t_{o} to t we obtain

∫_{t_{o}}^{t} ρ(s)q(s)ds ≤ ω(t_{o}) − ω(t) − ∫_{t_{o}}^{t} [\frac{Mω^{2}(s)}{ρ(s)r(s)} − (\frac{ρ̇(s)}{ρ(s)} − p(s)) ω(s)] ds,

Thus, for every t ≥ t_{o} we have

∫_{t_{o}}^{t} [ρ(s)q(s) − \frac{r(s)ρ(s)}{4M} (\frac{ρ̇(s)}{ρ(s)} − p(s))^{2}] ds ≤ ω(t_{o}) − ω(t)

− ∫_{t_{o}}^{t} [\sqrt{\frac{M}{ρ(s)r(s)}}ω(s) − \frac{\sqrt{r(s)ρ(s)}}{2\sqrt{M}} (\frac{ρ̇(s)}{ρ(s)} − p(s))]^{2} ds.

Hence, for all t ≥ t_{o} we have

∫_{t_{o}}^{t} [ρ(s)q(s) − \frac{r(s)ρ(s)}{4M} (\frac{ρ̇(s)}{ρ(s)} − p(s))^{2}] ds ≤ ω(t_{o}) − ω(t),

or

∫_{t_{o}}^{t} [ρ(s)q(s) − \frac{r(s)ρ(s)}{4M} (\frac{ρ̇(s)}{ρ(s)} − p(s))^{2}] ds ≤ ω(t_{o}) − ρ(t)\frac{r(t)ψ(x)f(ẋ)}{g(x(t))}.

From (4) we obtain

∫_{t_{o}}^{t} [ρ(s)q(s) − \frac{r(s)ρ(s)}{4M} (\frac{ρ̇(s)}{ρ(s)} − p(s))^{2}] ds ≤ ω(t_{o}) − L_{1}ρ(t)\frac{r(t)ψ(x)ẋ}{g(x(t))}.

Integrate again from t_{0} to t we obtain

∫_{t_{o}}^{t} ∫_{t_{o}}^{s} [ρ(u)q(u) − \frac{r(u)ρ(u)}{4M} (\frac{ρ̇(u)}{ρ(u)} − p(u))^{2}] duds

≤ ω(t_{o})(t − t_{o}) − L_{1} ∫_{t_{o}}^{t} ρ(s)r(s)\frac{ψ(x)ẋ}{g(x)}ds.  (6)

Since r(s)ρ(s) is nonincreasing, then by the Bonnet’s theorem there exists a η ∈ [t_{o}, t] such that

−L_{1} ∫_{t_{o}}^{t} r(s)ρ(s)\frac{ψ(x)ẋ}{g(x(s))}ds = −L_{1}r(t_{o})ρ(t_{o}) ∫_{t_{o}}^{η} \frac{ψ(x)ẋ}{g(x(s))}ds
= L_{1}r(t_{o})ρ(t_{o}) ∫_{x(η)}^{x(t_{o})} \frac{ψ(y)}{g(y)}dy
< \{ \begin{matrix}0  if x(t_{o}) < x(η), \\ L_{1}r(t_{o})ρ(t_{o}) ∫_{ε}^{∞} \frac{ψ(y)}{g(y)}dy if x(t_{o}) > x(η), \end{matrix}

EJQTDE, 2007 No. 25, p. 4 hence

−∞ < −L_{1} ∫_{t_{o}}^{t} r(s)ρ(s)\frac{ψ(x)ẋ}{g(x(s))}ds < k_{1},

where

k_{1} = L_{1}r(t_{o})ρ(t_{o}) ∫_{ε}^{∞} \frac{ψ(y)}{g(y)}dy.

Hence (6) becomes

∫_{t_{o}}^{t} ∫_{t_{o}}^{s} [ρ(u)q(u) − \frac{r(u)ρ(u)}{4M} (\frac{ρ̇(u)}{ρ(u)} − p(u))^{2}] duds ≤ ω(t_{o})(t − t_{o}) + k_{1}. (7)

Divide (7) by t and take the upper limit as t → ∞ which contradicts the assumption (5). This completes the proof. Corollary 1. If the condition (5) in the above theorem is replaced by

lim_{t→∞} sup \frac{1}{t} ∫_{t_{o}}^{t} ∫_{t_{o}}^{s} r(u)ρ(u) (\frac{ρ̇(u)}{ρ(u)} − p(u))^{2} duds < ∞,

lim_{t→∞} sup \frac{1}{t} ∫_{t_{o}}^{t} ∫_{t_{o}}^{s} ρ(u)q(u)duds = ∞,

then the conclusion of theorem 1 still true. Remark 1. If p(t) ≡ 0, r(t) ≡ 1 and ρ(t) ≡ 1, then Theorem 1 reduce to Wintner theorem in [32]. Theorem 2. Suppose, in addition to conditions (1) , (2), (3) and (4), that there exist a positive function ρ ∈ C^{1}[t_{o}, ∞) such that

(r(t)ρ(t))^{·} ≥ 0, ((r(t)ρ(t))^{··} ≤ 0,
γ(t) = (r(t)p(t)ρ(t) − ρ̇(t)r(t)) ≥ 0 and γ̇(t) ≤ 0 for all t ≥ t_{o},  (8)

and

lim_{t→∞} inf ∫_{t_{o}}^{t} ρ(s)q(s)ds > −∞  (9)

hold. Then equation (E) is oscillatory if

lim_{t→∞} sup \frac{1}{t} ∫_{t_{o}}^{t} [∫_{t_{o}}^{s} ρ(u)q(u)du]^{2} ds = ∞.  (10)

Proof. On the contrary we assume that (E) has a nonoscillatory solution x(t). We suppose without loss of generality that x(t) > 0 for all t ∈ [t_{o}, ∞). We define the function ω(t) as

ω(t) = ρ(t)\frac{r(t)ψ(x(t))f(ẋ(t))}{g(x(t))} for all t ≥ t_{o}.

This and equation (E) imply

ω̇(t) ≤ \frac{ρ̇(t)}{ρ(t)}ω(t) − ρ(t)[q(t) + p(t)φ(1, \frac{ω(t)}{ρ(t)})] − ρ(t)\frac{r(t)ψ(x(t))f(ẋ)g′(x(t))ẋ(t)}{g^{2}(x(t))}.

From (1), (2) and (4) we obtain

ρ(t)q(t) ≤ −ω̇(t) + \frac{ρ̇(t)}{ρ(t)}ω(t) − p(t)ω(t) − M \frac{1}{ρ(t)r(t)}ω^{2}(t),  (11)

or

ρ(t)q(t) ≤ −ω̇(t) − γ(t)\frac{ψ(x)f(ẋ)}{g(x(t))} − M \frac{1}{ρ(t)r(t)}ω^{2}(t).

From (4) we have

ρ(t)q(t) ≤ −ω̇(t) − L_{1}γ(t)\frac{ψ(x)ẋ}{g(x(t))} − M \frac{1}{ρ(t)r(t)}ω^{2}(t).  (12)

Integrating from T to t we obtain

∫_{T}^{t} ρ(s)q(s)ds ≤ ω(T) − ω(t) − L_{1} ∫_{T}^{t} γ(s)\frac{ψ(x)ẋ}{g(x(s))}ds − M ∫_{T}^{t} \frac{1}{ρ(s)r(s)}ω^{2}(s)ds.

(13)

Now evaluate the integral

− ∫_{T}^{t} γ(s)\frac{ψ(x(s))ẋ(s)}{g(x(s))}ds.

Since γ(t) is nonincreasing, then by the Bonnet’s theorem there exists a η ∈ [T, t] such that

− ∫_{T}^{t} γ(s)\frac{ψ(x(s))ẋ(s)}{g(x(s))}ds = −γ(T) ∫_{T}^{η} \frac{ψ(x(s))ẋ(s)}{g(x(s))}ds

= −γ(T) ∫_{x(T)}^{x(η)} \frac{ψ(y)}{g(y)}dy

= γ(T) ∫_{x(η)}^{x(T)} \frac{ψ(y)}{g(y)}dy

< \{ \begin{matrix}0 if x(η) > x(T), \\ γ(T) ∫_{ε}^{∞} \frac{ψ(y)}{g(y)}dy if x(η) < x(T), \end{matrix}

hence

−∞ < − ∫_{T}^{t} γ(s)\frac{ψ(x)ẋ}{g(x(s))}ds ≤ k_{2},  (14)

where

k_{2} = γ(T) ∫_{ε}^{∞} \frac{ψ(y)}{g(y)}dy.

From (14) in (13), we have

∫_{T}^{t} ρ(s)q(s)ds ≤ ω(T) − ω(t) + L_{1}k_{2} − M ∫_{T}^{t} \frac{1}{ρ(s)r(s)}ω^{2}(s)ds. (15)

Thus, we obtain for t ≥ T ≥ t_{o}

∫_{t_{o}}^{t} ρ(s)q(s)ds = ∫_{t_{o}}^{T} ρ(s)q(s)ds + ∫_{T}^{t} ρ(s)q(s)ds
≤ C_{1} − ω(t) − M ∫_{T}^{t} \frac{ω^{2}(s)}{ρ(s)r(s)}ds,  (16)

where

C_{1} = ω(T) + L_{1}k_{2} + ∫_{t_{o}}^{T} ρ(s)q(s)ds.

We consider the following two cases: Case 1. The integral

∫_{T}^{∞} \frac{ω^{2}(s)}{ρ(s)r(s)}ds is finite.

Thus, there exists a positive constant N such that

∫_{T}^{t} \frac{ω^{2}(s)}{ρ(s)r(s)}ds ≤ N for all t ≥ T.  (17)

Thus, we obtain for t ≥ T

[∫_{t_{o}}^{t} ρ(s)q(s)ds]^{2} ≤ \{C_{1} − ω(t) − M ∫_{T}^{t} \frac{ω^{2}(s)}{ρ(s)r(s)}ds\}^{2}

≤ 4C_{1}^{2} + 4ω^{2}(t) + 4M^{2} [∫_{T}^{t} \frac{ω^{2}(s)}{ρ(s)r(s)}ds]^{2} .

Therefore, by taking into account (17), we conclude that

[∫_{t_{o}}^{t} ρ(s)q(s)ds]^{2} ≤ C_{2} + 4ω^{2}(t),

where

C_{2} = 4C_{1}^{2} + 4M^{2} [∫_{T}^{t} \frac{ω^{2}(s)}{ρ(s)r(s)}ds]^{2} .

Thus, we obtain for every t ≥ T

∫_{t_{o}}^{t} [∫_{t_{o}}^{s} ρ(u)q(u)du]^{2} ds = ∫_{t_{o}}^{T} [∫_{t_{o}}^{s} ρ(u)q(u)du]^{2} ds

+ ∫_{T}^{t} [∫_{t_{o}}^{s} ρ(u)q(u)du]^{2} ds

= C_{3} + ∫_{T}^{t} [∫_{t_{o}}^{s} ρ(u)q(u)du]^{2} ds
≤ C_{3} + C_{2}(t − T) + 4 ∫_{T}^{t} ω^{2}(s)ds
= C_{3} + C_{2}(t − T) + 4 ∫_{T}^{t} ρ(s)r(s)\frac{ω^{2}(s)}{ρ(s)r(s)}ds.

Since ρ(t)r(t) is positive and nondecreasing for t ∈ [t_{o}, ∞), the Bonnet’s theorem would ensures the existence of T_{1} ∈ [T, t] such that

∫_{T}^{t} ρ(s)r(s)\frac{ω^{2}(s)}{ρ(s)r(s)}ds = ρ(t)r(t) ∫_{T_{1}}^{t} \frac{ω^{2}(s)}{ρ(s)r(s)}ds.

Also, since ρ(t)r(t) is positive on [t_{o}, ∞) and (ρ(t)r(t))^{·} is nonnegative and bounded above, it follow that ρ(t)r(t) ≤ βt for all large t where β > 0 is a constant. Which ensures

∫_{t_{o}}^{∞} \frac{ds}{ρ(s)r(s)} = ∞.

Thus, we conclude that

∫_{t_{o}}^{t} [∫_{t_{o}}^{s} ρ(u)q(u)du]^{2} ds ≤ C_{3} + C_{2}(t − T) + 4βt ∫_{T}^{t} \frac{ω^{2}(s)}{ρ(s)r(s)}ds.

So, we have

∫_{t_{o}}^{t} [∫_{t_{o}}^{s} ρ(u)q(u)du]^{2} ds ≤ C_{3} + C_{2}(t − T) + 4Nβt.

This implies

lim_{t→∞} sup \frac{1}{t} ∫_{t_{o}}^{t} [∫_{t_{o}}^{s} ρ(u)q(u)du]^{2} ≤ C_{2} + 4Nβ < ∞,

which contradicts (10). Case 2. The integral

∫_{T}^{∞} \frac{Kω^{2}(s)}{ρ(s)r(s)}ds is infinite.  (18)

By condition (2) , we see that

∫_{T}^{∞} \frac{g′(x)}{ρ(s)r(s)ψ(x)}ω^{2}(s)ds = ∞.

By virtue of condition (9), it follow from (16) for constant B

−ω(t) ≥ B + \frac{1}{L_{2}} ∫_{T}^{t} \frac{g′(x(s))}{ρ(s)r(s)ψ(x(s))}ω^{2}(s)ds for every t ≥ T. (19)

We can consider a T_{2} ≥ T such that

C = B + \frac{1}{L_{2}} ∫_{T}^{T_{2}} \frac{g′(x(s))}{ρ(s)r(s)ψ(x(s))}ω^{2}(s)ds > 0.

Then (19)ensures that w(t) is negative on [T_{2}, ∞). Now, (19) gives

\frac{1}{L_{2}} \frac{g′(x)ω^{2}(t)}{ρ(t)r(t)ψ(x)} \{B + \frac{1}{L_{2}} ∫_{T}^{t} \frac{g′(x(s))}{ρ(s)r(s)ψ(x)}ω^{2}(s)ds\}^{−1} ≥ \frac{−g′(x(t))f(ẋ(t))}{L_{2}g(x(t))}

≥ \frac{−g′(x(t))ẋ(t)}{g(x(t))},

and consequently for all t ≥ T_{2}

log \frac{B + \frac{1}{L_{2}} ∫_{T}^{t} \frac{g′(x)}{ρ(s)r(s)ψ(x)}ω^{2}(s)ds}{C} ≥ log \frac{g(x(T_{2}))}{g(x(t))} for t ≥ T_{2}.

Hence

B + \frac{1}{L_{2}} ∫_{T}^{t} \frac{g′(x)}{ρ(s)r(s)ψ(x)}ω^{2}(s)ds ≥ C \frac{g(x(T))}{g(x(t))}.

So, (19) yields

−ω(t) ≥ \frac{C′}{g(x(t))},

where C′ = Cg(x(T)). Thus, we have

ψ(x)f(ẋ) ≤ −C′ \frac{1}{ρ(t)r(t)}.

From (4) we have

ψ(x)ẋ ≤ \frac{−C′}{L_{1}} \frac{1}{ρ(t)r(t)}.

Integrate from T_{2} to t, we have

∫_{x(T)}^{x(t)} ψ(y)dy ≤ \frac{−C′}{L_{1}} ∫_{T}^{t} \frac{1}{ρ(s)r(s)}ds → −∞ as t → ∞,

a contradiction to the fact x(t) > 0 for t ≥ t_{o}, and hence (18) fails. This completes the proof. Remark 2. If p(t) ≡ 0 and f(ẋ) = ẋ then Theorem 2 reduce to Theorem 1 of Elabbasy [3]. Theorem 3. Suppose, in addition to (1) , (2) and

f^{2}(y) ≤ Lyf(y),  (20)

that there exist a positive function ρ ∈ C^{1}[t_{o}, ∞). Moreover, assume that there exist continuous functions H, h : D ≡ {(t, s), t ≥ s ≥ t_{o}} → R and H has a continuous and nonpositive partial derivative on D with respect to the second variable such that

H(t, t) = 0 for all t ≥ t_{o}, H(t, s) > 0 for all t > s ≥ t_{o},
−\frac{∂H(t, s)}{∂s} = h(t, s)\sqrt{H(t, s)} for all (t, s) ∈ D.

Then equation (E) is oscillatory if

lim_{t→∞} sup \frac{1}{H(t, t_{o})} ∫_{t_{o}}^{t} \{H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)\} ds = ∞. (21)

Where

Q(t, s) = [h(t, s) − (\frac{ρ̇(s)}{ρ(s)} − p(s)) \sqrt{H(t, s)}] and M = \frac{K}{L} .

Proof. On the contrary we assume that (E) has a nonoscillatory solution x(t). We suppose without loss of generality that x(t) > 0 for all t ∈ [t_{o}, ∞). We define the function ω(t) as

ω(t) = ρ(t)\frac{r(t)ψ(x(t))f(ẋ(t))}{g(x(t))} for all t ≥ t_{o}.

This and equation (E) imply

ω̇(t) ≤ \frac{ρ̇(t)}{ρ(t)}ω(t)−ρ(t)[q(t)+p(t)φ(1, \frac{ω(t)}{ρ(t)})]−ρ(t)\frac{r(t)ψ(x(t))f(ẋ(t))g′(x(t))ẋ(t)}{g^{2}(x(t))}.

From (1), (2) and (20) we obtain

ω̇(t) ≤ (\frac{ρ̇(t)}{ρ(t)} − p(s)) ω(t) − ρ(t)q(t) − \frac{M}{ρ(t)r(t)}ω^{2}(t).

Thus, for every t ≥ T, we have

∫_{T}^{t} H(t, s)ρ(s)q(s)ds ≤ ∫_{T}^{t} (\frac{ρ̇(t)}{ρ(t)} − p(s)) H(t, s)ω(s)ds

− ∫_{T}^{t} H(t, s)ω̇(s)ds − M ∫_{T}^{t} \frac{H(t, s)}{ρ(s)r(s)}ω^{2}(s)ds.

Since

− ∫_{T}^{t} H(t, s)ω̇(s)ds = H(t, T)ω(T) + ∫_{T}^{t} \frac{∂H(t, s)}{∂s}ω(s)ds
= H(t, T)ω(T) − ∫_{T}^{t} h(t, s)\sqrt{H(t, s)}ω(s)ds.

The previous inequality becomes

∫_{T}^{t} H(t, s)ρ(s)q(s)ds ≤ H(t, T)ω(T) − ∫_{T}^{t} Q(t, s)\sqrt{H(t, s)}ω(s)ds

−M ∫_{T}^{t} \frac{H(t, s)}{ρ(s)r(s)}ω^{2}(s)ds.

Hence we have

∫_{T}^{t} [H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)] ds ≤ H(t, T)ω(T)

−∫_{T}^{t} (\begin{matrix}\sqrt{M} & \sqrt{H(t, s)} \\ √ & \overline{r(s)ρ(s)} \end{matrix}ω(s) + \frac{\sqrt{r(s)ρ(s)}}{2\sqrt{M}}Q(t, s))^{2} ds. (22)

By (22) we have for every t ≥ T ≥ t_{o}

∫_{T}^{t} [H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)] ds ≤ H(t, T)ω(T)  (23)
≤ H(t, T) |ω(T)|
≤ H(t, t_{o}) |ω(T)| .

We use the above inequality for T = T_{o} to obtain

∫_{T_{o}}^{t} [H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)] ds ≤ H(t, t_{o}) |ω(T_{o})| .

Therefore,

∫_{t_{o}}^{t} [H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)] ds

= ∫_{t_{o}}^{T_{o}} [H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)] ds

+ ∫_{T_{o}}^{t} [H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)] ds.

Hence for every t ≥ t_{o} we have

∫_{t_{o}}^{t} [H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)] ds

≤ H(t, t_{o}) {∫_{t_{o}}^{T_{o}} ρ(s) |q(s)| ds + |ω(T_{o})|} .  (24)

Dividing (24) by H(t, t_{o}) and take the upper limit as t → ∞ we get

lim_{t→∞} sup \frac{1}{H(t, t_{o})} ∫_{t_{o}}^{t} [H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)] ds < ∞,

which contradicts (21). This completes the proof. Corollary 2. If condition (21) in Theorem 3 is replaced by

lim_{t→∞} sup \frac{1}{H(t, t_{o})} ∫_{t_{o}}^{t} H(t, s)ρ(s)q(s)ds = ∞,

lim_{t→∞} sup \frac{1}{H(t, t_{o})} ∫_{t_{o}}^{t} r(s)ρ(s)Q^{2}(t, s)ds < ∞,

then the conclusion of Theorem 3 remains valid. Corollary 3. If we take H(t, s) = (t − s)^{α} for α ≥ 2, then the condition (21) in the above theorem becomes

lim_{t→∞} sup \frac{1}{t^{α}} ∫_{t_{o}}^{t}(t − s)^{α}ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}

× [α − (t − s) (\frac{ρ^{·}(s)}{ρ(s)} − p(s))]^{2} (t − s)^{α−2}ds = ∞.

Remark 3. Theorem 3 extended and improved Theorem 2 in [4]. Remark 4. If f(x^{·}) = x^{·}, ψ(x) = 1 and φ (g(x), r(t)ψ(x)f(x^{·})) = x^{·}, then Corollary 2 extend Nagabuchi and Yamamoto theorem in [20] and Yan Theorem with g(x) = x in [35]. Example 1. Consider the differential equation

(t\frac{4t^{2} + 2 cos^{2}(ln t)}{t^{2} + 2 cos^{2}(ln t)}ẋ(t) (\frac{1 + 2ẋ^{2}(t)}{4 + 2ẋ^{2}(t)}))^{·} + \frac{sin^{2}(ln t)}{t^{2}}

× [\frac{(t\frac{4t^{2}+2 cos^{2}(ln t)}{t^{2}+2 cos^{2}(ln t)} ẋ(t) (\frac{1+2ẋ^{2}(t)}{4+2ẋ^{2}(t)}))^{2} + x^{2}(t)}{x(t)}] + \frac{1}{2t}x(t) = 0, t ≥ t_{o} = 1 .

If we take ρ(t) = 1 and H(t, s) = (t − s)^{2}, then we see that all hypotheses of Theorem 3 are satisfied where

lim_{t→∞} sup \frac{1}{H(t, t_{o})} ∫_{t_{o}}^{t} \{H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)\} ds

= lim_{t→∞} sup \frac{1}{(t − 1)^{2}} ∫_{1}^{t} [\frac{(t − s)^{2}}{2s} − \frac{s}{4} \frac{4s^{2} + 2 cos^{2}(ln s)}{s^{2} + 2 cos^{2}(ln s)}

× (2 + \frac{(t − s) sin^{2} (ln s)}{2s})^{2}] ds

≥ lim_{t→∞} sup \frac{1}{(t − 1)^{2}} ∫_{1}^{t} [\frac{(t − s)^{2}}{2s} − \frac{s}{4} [2 + \frac{(t − s)}{2s}]^{2}] ds

= lim_{t→∞} sup \frac{1}{(t − 1)^{2}} (\frac{11}{8}t − \frac{45}{32}t^{2} + \frac{7}{16}t^{2} ln t + \frac{1}{32}) = ∞

Hence, this equation is oscillatory by theorem 3. One such solution of this equation is x(t) = sin(ln t). Theorem 4. Suppose, in addition to condition (1), (2) and (20), that there exist a positive function ρ ∈ C^{1}[t_{o}, ∞). Moreover, assume that the function H and h be as in Theorem 3, and let

0 < inf_{s ≥ t_{o}} [ lim_{t→∞} inf \frac{H(t, s)}{H(t, t_{o})}] ≤ ∞.  (25)

Suppose that there exists a function Ω ∈ C([t_{o}, ∞), R) such that

lim_{t→∞} sup \frac{1}{H(t, t_{o})} ∫_{t_{o}}^{t} ρ(s)r(s)Q^{2}(t, s)ds < ∞,  (26)

lim_{t→∞} sup \frac{1}{H(t, T)} ∫_{T}^{t}\{H(t, s)ρ(s)q(s) − \frac{ρ(s)r(s)Q^{2}(t, s)}{4M}\}ds ≥ Ω(T), (27)

for every T ≥ t_{o}. Then equation (E) is oscillatory if

∫_{t_{o}}^{∞} \frac{Ω_{+}^{2}(s)}{ρ(s)r(s)} = ∞,  (28)

where Ω_{+}(t) =max{Ω(t), 0} for t ≥ t_{o}. Proof. On the contrary we assume that (E) has a nonoscillatory solution x(t). We suppose without loss of generality that x(t) > 0 for all t ∈ [t_{o}, ∞). Defining ω(t) as in the proof of Theorem 3, we obtain (22) and hence for t > T ≥ t_{o}, we get

lim_{t→∞} sup \frac{1}{H(t, T)} ∫_{T}^{t} [H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)] ds

≤ ω(T) − lim_{t→∞} inf \frac{1}{H(t, T)} ∫_{T}^{t} [\begin{matrix}\sqrt{M} & \sqrt{H(t, s)} \\ √ & \overline{r(s)ρ(s)} \end{matrix}ω(s) + \frac{\sqrt{r(s)ρ(s)}}{2\sqrt{M}}Q(t, s)]^{2} ds.

Thus, by condition (27) we have for T ≥ t_{o}

ω(T) ≥ Ω(T)+ lim_{t→∞} inf \frac{1}{H(t, T)} ∫_{T}^{t} [\begin{matrix}\sqrt{M} & \sqrt{H(t, s)} \\ √ & \overline{r(s)ρ(s)} \end{matrix}ω(s) + \frac{\sqrt{r(s)ρ(s)}}{2\sqrt{M}}Q(t, s)]^{2} ds.

This shows that

ω(T) ≥ Ω(T),  (29)

and

lim_{t→∞} inf \frac{1}{H(t, T)} ∫_{T}^{t} [\begin{matrix}\sqrt{M} & \sqrt{H(t, s)} \\ √ & \overline{r(s)ρ(s)} \end{matrix}ω(s) + \frac{\sqrt{r(s)ρ(s)}}{2\sqrt{M}}Q(t, s)]^{2} ds < ∞.

Hence

∞ > lim_{t→∞} inf \frac{1}{H(t, t_{o})} ∫_{t_{o}}^{t} [\begin{matrix}\sqrt{M} & \sqrt{H(t, s)} \\ √ & \overline{r(s)ρ(s)} \end{matrix}ω(s) + \frac{\sqrt{r(s)ρ(s)}}{2\sqrt{M}}Q(t, s)]^{2} ds

≥ lim_{t→∞} inf \frac{1}{H(t, t_{o})} ∫_{t_{o}}^{t} [\frac{M H(t, s)}{r(s)ρ(s)}ω^{2}(s) + Q(t, s)\sqrt{H(t, s)} ω(s)] ds. (30)

Define the function α(t) and β(t) as follows

α(t) = \frac{1}{H(t, t_{o})} ∫_{t_{o}}^{t} \frac{M H(t, s)}{r(s)ρ(s)}ω^{2}(s)ds,

β(t) = \frac{1}{H(t, t_{o})} ∫_{t_{o}}^{t} Q(t, s)\sqrt{H(t, s)} ω(s)ds.

Then (30) may be written as

lim_{t→∞} inf [α(t) + β(t)] < ∞.  (31)

In order to show that

∫_{t_{o}}^{∞} \frac{ω^{2}(s)}{ρ(s)r(s)} < ∞.  (32)

Now, suppose that

∫_{t_{o}}^{∞} \frac{ω^{2}(s)}{ρ(s)r(s)} = ∞.  (33)

By (25) we can easily see that

lim_{t→∞}α(t) = ∞.  (34)

Next, let us consider a sequence {T_{n}}_{n=1,2,3,....}in (t_{o}, ∞) with lim_{n→∞}T_{n} = ∞ and such that

lim_{n→∞} [α(T_{n}) + β(T_{n})] = lim_{t→∞} inf [α(t) + β(t)] .

EJQTDE, 2007 No. 25, p. 14 Now, by (31), there exists a constant N such that

α(T_{n}) + β(T_{n}) ≤ N  (n = 1, 2, ...).  (35)

Furthermore (34) guarantees that

lim_{n→∞}α(T_{n}) = ∞,  (36)

and hence (35) gives

lim_{n→∞}β(T_{n}) = −∞.  (37)

By taking into account (36), from (35), we derive

1 + \frac{β(T_{n})}{α(T_{n})} ≤ \frac{N}{α(T_{n})} < \frac{1}{2},

provided that n is sufficiently large. Thus,

\frac{β(T_{n})}{α(T_{n})} < \frac{−1}{2} for all large n,

which by (37), ensures that

lim_{n→∞} \frac{β^{2}(T_{n})}{α(T_{n})} = ∞.  (38)

On the other hand, by Schawrz inequality, we have for any positive integer n,

β^{2}(T_{n}) = \frac{1}{H^{2}(T_{n}, t_{o})} [∫_{t_{o}}^{T_{n}} Q(T_{n}, s)\sqrt{H(T_{n}, s)} ω(s)ds]^{2}

≤ [\frac{1}{H(T_{n}, t_{o})} ∫_{t_{o}}^{T_{n}} \frac{M H(T_{n}, s)}{r(s)ρ(s)}ω^{2}(s)ds]

× [\frac{1}{H(T_{n}, t_{o})} ∫_{t_{o}}^{T_{n}} \frac{r(s)ρ(s)}{M}Q^{2}(T_{n}, s)ds]

= α(T_{n}) [\frac{1}{H(T_{n}, t_{o})} ∫_{t_{o}}^{T_{n}} \frac{r(s)ρ(s)}{M}Q^{2}(T_{n}, s)ds] ,

or

\frac{β^{2}(T_{n})}{α(T_{n})} ≤ \frac{1}{H(T_{n}, t_{o})} ∫_{t_{o}}^{T_{n}} \frac{r(s)ρ(s)}{M}Q^{2}(T_{n}, s)ds.

It follow from (38) that

lim_{n→∞} \frac{1}{H(T_{n}, t_{o})} ∫_{t_{o}}^{T_{n}} r(s)ρ(s)Q^{2}(T_{n}, s)ds = ∞.  (39)

Consequently,

lim_{t→∞} sup \frac{1}{H(t, t_{o})} ∫_{t_{o}}^{t} ρ(s)r(s)Q^{2}(t, s)ds = ∞,

but the latter contradicts the assumption (26) . Hence, (33) fails to hold. Consequently, we have proved that inequality (32) holds. Finally, by (29) we obtain

∫_{t_{o}}^{∞} \frac{Ω_{+}^{2}(s)}{ρ(s)r(s)} ≤ ∫_{t_{o}}^{∞} \frac{ω^{2}(s)}{ρ(s)r(s)} < ∞,

which contradicts the assumption (28). Therefore, equation (E) is oscillatory. Example 2. Consider the differential equation

(\frac{1 + sin^{2} t}{t^{2}} \frac{1}{1 + x^{2}} ẋ)^{·} + \frac{2}{t^{3}} ẋ

+ \frac{1 + sin^{2} t}{t^{2}(3 + 2 sin^{2} t)}x [2 + \frac{1}{1 + x^{2}} ] = 0, t ≥ t_{o} = 1. (40)

We note that

\frac{g′(x)}{ψ(x)} = 3 + \frac{2x^{4}}{(1 + x^{2})} ≥ 3 = K.

If we take ρ(t) = 1 and H(t, s) = (t − s)^{2}, then we have

lim_{t→∞} sup \frac{1}{H(t, t_{o})} ∫_{1}^{t} r(s)ρ(s)Q^{2}(t, s)ds

= lim_{t→∞} sup \frac{1}{(t − 1)^{2}} ∫_{1}^{t} \frac{(1 + sin^{2} s)}{s^{2}} (2 + \frac{2t}{s^{3}} − \frac{2}{s^{2}} )^{2} ds

≤ lim_{t→∞} sup \frac{1}{(t − 1)^{2}} ∫_{1}^{t} \frac{2}{s^{2}} (2 + \frac{2t}{s^{3}} − \frac{2}{s^{2}} )^{2} ds = \frac{8}{7} < ∞,

lim_{t→∞} sup \frac{1}{H(t, T)} ∫_{T}^{t} (H(t, s)ρ(s)q(s) − \frac{r(s)ρ(s)}{4M}Q^{2}(t, s)) ds

= lim_{t→∞} sup \frac{1}{(t − T)^{2}} ∫_{T}^{t} \frac{(t − s)^{2} (1 + sin^{2} s)}{s^{2}(3 + 2 sin^{2} s)} − \frac{(1 + sin^{2} s)}{12s^{2}} (2 + \frac{2t}{s^{3}} − \frac{2}{s^{2}} )^{2} ds

≥ lim_{t→∞} sup \frac{1}{(t − T)^{2}} ∫_{T}^{t} ((t − s)^{2}(\frac{1}{3s^{2}} ) − \frac{1}{6s^{2}} (2 + \frac{2t}{s^{3}} − \frac{2}{s^{2}} )^{2}) ds

= \frac{1}{21} \frac{7T^{6} − 2}{T^{7}} =^{def} Ω(T),

and, finally,

∫_{1}^{∞} \frac{Ω_{+}^{2}(s)}{ρ(s)r(s)}ds = ∫_{1}^{∞} (\frac{1}{21} \frac{7s^{6} − 2}{s^{7}})^{2} (\frac{s^{2}}{1 + sin^{2} s}) ds

≥ ∫_{1}^{t} (\frac{1}{21} \frac{7s^{6} − 2}{s^{7}})^{2} (\frac{s^{2}}{2} ) ds = ∞.

Thus we conclude that (40) is oscillatory by Theorem 4. As a matter of fact, x(t) = sin t is an oscillatory solution of this equation.

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(Received September 19, 2006)